The set of equations (2)-(4) can therefore be seen as a model for the evolution of a two-phase flow consisting of two immiscible incompressible fluids with constant densities ρw and ρa and different viscosity coefficients µw and µa. In this
respect, in view of the numerical simulation, we could regard equation (2) as the candidate for updating the (unknown) interface location Γ , then treat equations (3)-(4) as a coupled system of Navier–Stokes equations in the two sub-domains
Ωw and Ωa: